A flexible one piece, insulative battery cover; battery cover assembly therewith; method of construction thereof, and method of enclosing a battery therewith are provided. The battery cover includes a flexible, tubular textile wall having a cavity extending between a bottom end and an open top end of the wall. The cover further includes a flexible textile lid having a free end and an opposite end attached to the wall via a hinge, with the lid being pivotal via the hinge between an open position, exposing the open top end of the wall, and a closed position, substantially closing off the top end. Further yet, the cover includes a fastener fixed to the lid adjacent the free end, with the fastener being configured to for releasable attachment to an elongate fastener member of the vehicle battery assembly.